Dyson-equation approach

We also want to mention that a Dyson-equation approach for propagators like the polarisation and the particle-pcurticle propagator has been formulated and used to derive a self-consistent extension of the RPA, also called cluster-Hartree-Fock approximation, that has been applied in the fields of plasma and nuclear physics [12-14], This formalism, however, has similar problems like Feshbach s theory and does not yield a universal, well-behaved optical potential because the two-particle space has to be restricted in order to make the approach well-defined [14]. 

For simplicity, the QD and SM indices in the e-ph constants have been omitted in Eq. (118) however, the frequencies and e-ph constants are obviously different in the both subsystems. In the proposed description, we assume that equilibrium Green s functions of the semiconductor and the quantum dot are known. However, to find QD equilibrium Green s function in a time-dependent field is not an easy task because it is not even clear whether Dyson equations for SM and QD Keldysh functions exist for different types of fermions interacting with each other. This problem is rather complicated even for molecular wires [54], Thus, we expect this problem to be even more complicated for solar cell systems where the interaction with light makes the problem essentially time dependent. In this section, we prove that Dyson equations for nonequilibrium Green s functions do exist. In our description, we adopt a graduate approach to the problem introducing different approximations step by step. As the first and the easiest step, we consider only uncorrelated electrons. 

To illustrate a FT, we use the Dyson equation that is characteristic of helds and we focus on the local density fluctuations that is the most natural variable in this approach. Two systems are considered a homogeneous Yukawa fluid and an ionic soluhon near a hard wall. 

The propagator [6-9] or Green s function method (GFM) [10-14] provides another approach to calculate the quasi-particle energy bands. The Dyson equation provides the exact E(N 1) energies in a formally one-particle picture, but the equation can only be solved approximately in real applications [57], With the irreducible self-energy part in the diagonal approximation being correct to second-order, the inverse Dyson equation can be written as [26]... 

Where does this fundamental difference to (2.27) come from To answer this question one has to remember that in the first two approaches some arbitrary orbital-dependent xc has been assumed, i.e. the form of E c has not been specified. On the other hand, in the present approach the use of the Dyson equation for both the KS and the interacting system automatically implies the use of the exact E c- In order to make closer contact between the first two and this third derivation, one thus has to study the exact xc in more detail. This will be the subject of Sect. 2.4. In the present section the comparison of (2.45) with (2.27) will for simplicity be restricted to the x-only limit, which corresponds to a lowest order expansion of E in the coupling constant e. In this limit the right-hand side of (2.45) reduces to... 

As in the STM case, the equations must in practice be solved numerically. Perturbation theory is seldom employed, and most calculations make a direct solution for the optical modes at a fixed frequency, either by a transfer matrix approach, or by using the Dyson equation to obtain the solution from that of an exactly soluble system (for example, free space). 

However, the domain of the QCD phase diagram where neutron star conditions are met is not yet accessible to Lattice QCD studies and theoretical approaches have to rely on nonperturbative QCD modeling. The class of models closest to QCD are Dyson-Schwinger equation (DSE) approaches which have been extended recently to finite temperatures and densities [11-13], Within simple, infrared-dominant DSE models early studies of quark stars [14] and diquark condensation [15] have been performed. 

Balabin lA, Onuchic JN. Connection between simple models and quantum chemical models for electron-transfer tunneling matrix element calculations A Dyson s equations-based approach. J. Phys. Chem. 1996 100 11573-11580. 

The size of the f a) subspace is quite large for reasonable basis sets. For N electrons and a spin orbital basis of rank K, there are K — N) K — 2)N/2 elements in the orthogonal complement space. The solution of the Dyson-like equation would require the inversion of a matrix of this dimension for every value of the energy parameter. This approximation of the self-energy first introduced by J. Linderberg and Y. Ohrn,, has been discussed by G. D. Purvis III, and Y. Ohrn, and J. Schirmer and L. S. Cederbaum . A simplified version of this approach is the so-called Diagonal 2p-h TDA , which becomes... 

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